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Chapter 10: Problem 49

Use Taylor's formula with \(n=2\) to find the quadratic approximation of \(f(x)=(1+x)^{k}\) at \(x=0\) ( \(k\) a constant). b. If \(k=3,\) for approximately what values of \(x\) in the interval [0,1] will the error in the quadratic approximation be less

Short Answer

Expert verified
The quadratic approximation for \((1+x)^k\) at \(x=0\) is \(1 + kx + \frac{k(k-1)}{2}x^2\). For \(k=3\), the approximation error is less than 0.01 when \(x \in [0, 0.1)\).

Step by step solution

01

Write Taylor's Formula

Taylor's formula for approximating a function near a point is given by: \[ f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 + R_2(x) \]where \(R_2(x)\) is the remainder term. For the quadratic approximation, we focus on the terms up to \(\frac{f''(a)}{2}(x-a)^2\).
02

Compute Derivatives

The function is \(f(x) = (1+x)^k\). First, find the first and second derivatives of this function:1. First derivative: \[f'(x) = k(1+x)^{k-1}\]2. Second derivative:\[f''(x) = k(k-1)(1+x)^{k-2}\]
03

Evaluate Derivatives at \(x = 0\)

Evaluate the function and its derivatives at \(x = 0\):1. \(f(0) = (1+0)^k = 1\)2. \(f'(0) = k(1+0)^{k-1} = k\)3. \(f''(0) = k(k-1)(1+0)^{k-2} = k(k-1)\)
04

Formulate the Quadratic Approximation

Substitute the evaluated derivatives into the Taylor's formula:\[f(x) \approx 1 + kx + \frac{k(k-1)}{2}x^2\]
05

Determine the Remainder Term in Approximation

The remainder term \(R_2(x)\) for Taylor's series up to \(n=2\) is given by:\[R_2(x) = \frac{f'''(\xi)}{6}x^3\]where \(\xi\) is some point in the interval [0,x]. The third derivative is needed for error approximation:
06

Find the Third Derivative

Calculate the third derivative:\[f'''(x) = k(k-1)(k-2)(1+x)^{k-3}\]
07

Error Estimation for Specific k

Substitute \(k=3\) into the quadratic approximation and third derivative:1. Approximation: \(f(x) \approx 1 + 3x + 3x^2\)2. Third derivative: \(f'''(x) = 6(1+x)^0 = 6\)Use the error formula:\[|R_2(x)| = \left| \frac{6}{6}x^3 \right| = |x^3|\]The error \(|x^3| < 0.01\) must hold for accurate approximation.
08

Solve the Error Inequality

Solve \(|x^3| < 0.01\) to determine values of \(x\):\[x^3 < 0.01\]Take cube root:\[x < 0.1\]Thus, \(x\) should be in \([0, 0.1)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Taylor's Formula
Taylor's formula is a method widely used in mathematics to approximate functions using polynomials. It is especially useful when you want to find an approximation of a function near a specific point, say "a". The general structure of Taylor's formula is as follows:
  • The function is approximated as a sum of its derivatives evaluated at that point "a" and multiplied by powers of the distance from "a", that is, \( (x-a) \).
  • Mathematically, the Taylor approximation of order n can be expressed as: \[ f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 + \ldots + R_n(x) \]
  • This series expands up to n terms, where each term involves a derivative of higher order.
In our case, since we're finding a quadratic approximation, we'll include terms up to \( f''(a) \), disregarding higher derivatives. This core idea facilitates approximating complex functions easily and is especially useful when direct computation of the function is difficult.
Error Estimation
Estimating the error in Taylor's polynomial approximation is crucial in understanding how accurate our approximation is. The error is represented as a remainder term, often denoted by \( R_n(x) \).
  • This tells us how much the polynomial approximation differs from the actual function value.
  • For \( n=2 \), the remainder \( R_2(x) \) is given by \[ R_2(x) = \frac{f'''(\xi)}{6}x^3 \], where \( \xi \) is a number between the approximation point "a" and "x".
  • In practical applications, you often estimate the value of \( \xi \) to find a bound on the error.
In our example, if we want the quadratic approximation with \( k=3 \) to be accurate within an error less than 0.01, we set the inequality to \( |x^3| < 0.01 \). Solving this lets us know the range of \( x \) values for which the approximation will be valid within the desired error margin.
Mathematical Derivatives
When dealing with Taylor's formulas, derivatives play a crucial role, as each term in the polynomial is a function of derivatives of different orders. Understanding derivatives is essential for constructing Taylor polynomials:
  • The first derivative, \( f'(x) \), gives the slope of the tangent line to the function and helps introduce linear terms to the approximation. \[ f'(x) = k(1+x)^{k-1} \]
  • The second derivative, \( f''(x) \), indicates the curvature of the function and allows for including quadratic terms. \[ f''(x) = k(k-1)(1+x)^{k-2} \]
  • Higher derivatives, although not directly used in quadratic approximations, are important for assessing error terms. Here, the third derivative is: \[ f'''(x) = k(k-1)(k-2)(1+x)^{k-3} \]
Knowing how to compute and apply these derivatives is key to accurately using Taylor's polynomial for function approximation. They provide not just the tools to construct approximations but also ensure insights into how function values change over intervals.

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